Abstract: This lecture is dedicated to the memory of Israel M. Gelfand (1913-2009), a great mathematician who had a crucial influence on many areas of 20th century mathematics. His active mathematical life spanned nearly 80 years. I was happy to know him for more than 50 years and to collaborate with him on several projects. Gelfand had an absolutely special style in mathematics and I want to discuss this style and some of his achievements. One of Gelfand's lessons was to think about mathematics in a general setting, but to explain it with examples. Following this advice I will focus on one of Gelfand's discoveries - integral geometry. It started almost exactly 50 years ago when Gelfand extracted from the representations of the Lorentz group a problem in geometric analysis that could be naturally generalized to a situation in which the group disappeared. Gelfand's dream was to discover a geometrical universe that encompassed not only semisimple Lie groups, but other important mathematical realities as well. We will discuss how much has been done in these 50 years and how far we are today from a realization of Gelfand's dream.